3.653 \(\int x \sqrt{1-x^2} \cos ^{-1}(x) \, dx\)

Optimal. Leaf size=30 \[ \frac{x^3}{9}-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-\frac{x}{3} \]

[Out]

-x/3 + x^3/9 - ((1 - x^2)^(3/2)*ArcCos[x])/3

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Rubi [A]  time = 0.0313971, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {4678} \[ \frac{x^3}{9}-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-\frac{x}{3} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[1 - x^2]*ArcCos[x],x]

[Out]

-x/3 + x^3/9 - ((1 - x^2)^(3/2)*ArcCos[x])/3

Rule 4678

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcCos[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x \sqrt{1-x^2} \cos ^{-1}(x) \, dx &=-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-\frac{1}{3} \int \left (1-x^2\right ) \, dx\\ &=-\frac{x}{3}+\frac{x^3}{9}-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0276791, size = 26, normalized size = 0.87 \[ \frac{1}{9} \left (x^3-3 \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-3 x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[1 - x^2]*ArcCos[x],x]

[Out]

(-3*x + x^3 - 3*(1 - x^2)^(3/2)*ArcCos[x])/9

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Maple [C]  time = 0.15, size = 134, normalized size = 4.5 \begin{align*} -{\frac{i+3\,\arccos \left ( x \right ) }{72} \left ( 4\,i{x}^{3}-4\,\sqrt{-{x}^{2}+1}{x}^{2}-3\,ix+\sqrt{-{x}^{2}+1} \right ) }+{\frac{\arccos \left ( x \right ) +i}{8} \left ( ix-\sqrt{-{x}^{2}+1} \right ) }-{\frac{\arccos \left ( x \right ) -i}{8} \left ( ix+\sqrt{-{x}^{2}+1} \right ) }+{\frac{-i+3\,\arccos \left ( x \right ) }{72} \left ( 4\,i{x}^{3}+4\,\sqrt{-{x}^{2}+1}{x}^{2}-3\,ix-\sqrt{-{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arccos(x)*(-x^2+1)^(1/2),x)

[Out]

-1/72*(I+3*arccos(x))*(4*I*x^3-4*(-x^2+1)^(1/2)*x^2-3*I*x+(-x^2+1)^(1/2))+1/8*(arccos(x)+I)*(I*x-(-x^2+1)^(1/2
))-1/8*(arccos(x)-I)*(I*x+(-x^2+1)^(1/2))+1/72*(-I+3*arccos(x))*(4*I*x^3+4*(-x^2+1)^(1/2)*x^2-3*I*x-(-x^2+1)^(
1/2))

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Maxima [A]  time = 1.41756, size = 30, normalized size = 1. \begin{align*} \frac{1}{9} \, x^{3} - \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \arccos \left (x\right ) - \frac{1}{3} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccos(x)*(-x^2+1)^(1/2),x, algorithm="maxima")

[Out]

1/9*x^3 - 1/3*(-x^2 + 1)^(3/2)*arccos(x) - 1/3*x

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Fricas [A]  time = 2.68538, size = 78, normalized size = 2.6 \begin{align*} \frac{1}{9} \, x^{3} + \frac{1}{3} \,{\left (x^{2} - 1\right )} \sqrt{-x^{2} + 1} \arccos \left (x\right ) - \frac{1}{3} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccos(x)*(-x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/9*x^3 + 1/3*(x^2 - 1)*sqrt(-x^2 + 1)*arccos(x) - 1/3*x

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Sympy [A]  time = 1.00505, size = 37, normalized size = 1.23 \begin{align*} \frac{x^{3}}{9} + \frac{x^{2} \sqrt{1 - x^{2}} \operatorname{acos}{\left (x \right )}}{3} - \frac{x}{3} - \frac{\sqrt{1 - x^{2}} \operatorname{acos}{\left (x \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*acos(x)*(-x**2+1)**(1/2),x)

[Out]

x**3/9 + x**2*sqrt(1 - x**2)*acos(x)/3 - x/3 - sqrt(1 - x**2)*acos(x)/3

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Giac [A]  time = 1.08687, size = 30, normalized size = 1. \begin{align*} \frac{1}{9} \, x^{3} - \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \arccos \left (x\right ) - \frac{1}{3} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccos(x)*(-x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/9*x^3 - 1/3*(-x^2 + 1)^(3/2)*arccos(x) - 1/3*x